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Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, –1) and (4, 3, –1).

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Let us denote the points as follows:

⇒ O = (0, 0, 0)

⇒ A = (2, 1, 1)

⇒ B = (3, 5, –1)

⇒ C = (4, 3, –1)

If two lines of direction ratios (a1, b1, c1) and (a2, b2, c2) are said to be perpendicular to each other. Then the following condition is need to be satisfied:

⇒ a1 . a2 + b1 . b2 + c1 . c2 = 0 ……(1)

Let us assume the direction ratios for line OA be (r1, r2, r3) and BC be (r4, r5, r6)

We know that direction ratios for a line passing through points (x1, y1, z1) and (x2, y2, z2) is (x2 – x1, y2 – y1, z2 – z1).

Let’s find the direction ratios for the line OA

⇒ (r1, r2, r3) = (2 – 0, 1 – 0, 1 – 0)

⇒ (r1, r2, r3) = (2, 1, 1)

Let’s find the direction ratios for the line BC

⇒ (r4, r5, r6) = (4 – 3, 3 – 5, – 1 – (– 1))

⇒ (r4, r5, r6) = (4 – 3, 3 – 5, – 1 + 1)

⇒ (r4, r5, r6) = (1, –2, 0)

Let us check whether the lines are perpendicular or not using (1)

⇒ r1 . r4 + r2 . r5 + r3 . r6 = (2 × 1) + (1 × –2) +(1 × 0)

⇒ r1 . r4 + r2 . r5 + r3 . r6 = 2 – 2 + 0

⇒ r1 . r4 + r2 . r5 + r3 . r6 = 0

Since the condition is clearly satisfied, we can say that the given lines are perpendicular to each other.

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